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| Mirrors > Home > MPE Home > Th. List > elin | Structured version Visualization version GIF version | ||
| Description: Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.) |
| Ref | Expression |
|---|---|
| elin | ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ V) | |
| 2 | elex 3501 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ V) |
| 4 | eleq1 2829 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
| 5 | eleq1 2829 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
| 6 | 4, 5 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) |
| 7 | df-in 3958 | . . 3 ⊢ (𝐵 ∩ 𝐶) = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)} | |
| 8 | 6, 7 | elab2g 3680 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) |
| 9 | 1, 3, 8 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
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